This pa­per con­sists of two parts. In the first part we ex­tend our earli­er res­ults [Stroock and Varadhan 1972] on the strong max­im­um prin­ciple to a broad­er class of op­er­at­ors, namely de­gen­er­ate para­bol­ic op­er­at­ors \( \partial/\partial t + L_t \), where
\[ L_t = \tfrac{1}{2}\nabla\cdot(a(t,x)\nabla)+b(t,x)\cdot\nabla \]
with \( a \) and \( b \) suit­ably smooth. This leads to a gen­er­al­iz­a­tion of the res­ults of M. Bony [1969] that was sought by C. D. Hill [1970]. It is also re­lated to a re­cent res­ult of M. Red­hef­fer [1971]. The second part of the pa­per is de­voted to the study of the first bound­ary value prob­lem for de­gen­er­ate el­lipt­ic op­er­at­ors
\[ L = \tfrac{1}{2}\nabla\cdot(a(x)\nabla) + b(x)\cdot\nabla - k(x) \]
in smooth re­gions.

The re­la­tion­ship between \( H^p(R^d) \), \( 1\leq p < \infty \), and the in­teg­rabil­ity of cer­tain func­tion­als of Browni­an mo­tion is es­tab­lished us­ing the con­nec­tion between prob­ab­il­ist­ic and ana­lyt­ic no­tions of func­tions with bounded mean os­cil­la­tion. An ap­plic­a­tion of this re­la­tion­ship is giv­en in the de­riv­a­tion of an in­ter­pol­a­tion the­or­em for op­er­at­ors tak­ing \( H^1(R^d) \) to \( L^1(R^d) \).

The Bib­li­o­graph­ic Data, be­ing a mat­ter of fact and
not cre­at­ive ex­pres­sion, is not sub­ject to copy­right.
To the ex­tent pos­sible un­der law,
Math­em­at­ic­al Sci­ences Pub­lish­ers
has waived all copy­right and re­lated or neigh­bor­ing rights to the
Bib­li­o­graph­ies on Cel­eb­ra­tio Math­em­at­ica,
in their par­tic­u­lar ex­pres­sion as text, HTML, Bib­TeX data or oth­er­wise.

The Ab­stracts of the bib­li­o­graph­ic items may be copy­righted ma­ter­i­al whose use has not
been spe­cific­ally au­thor­ized by the copy­right own­er.
We be­lieve that this not-for-profit, edu­ca­tion­al use con­sti­tutes a fair use of the
copy­righted ma­ter­i­al,
as provided for in Sec­tion 107 of the U.S. Copy­right Law. If you wish to use this copy­righted ma­ter­i­al for
pur­poses that go bey­ond fair use, you must ob­tain per­mis­sion from the copy­right own­er.